### Uniqueness and nondegeneracy of ground states for non-local equations in 1D

Frank, R *(Princeton University)*

Thursday 02 August 2012, 14:00-14:45

Seminar Room 1, Newton Institute

#### Abstract

We prove uniqueness of energy minimizing solutions Q for the nonlinear equation (-\Delta)^s Q + Q - Q^{\alpha+1}= 0 in 1D, where 0 < s < 1 and 0 < \alpha < \frac{4s}{1-2s} for s < 1/2 and 0 < \alpha < \infty for s \geq 1/2. Here (-\Delta)^s is the fractional Laplacian. As a technical key result, we show that the associated linearized operator is nondegenerate, in the sense that its kernel is spanned by Q'. This solves an open problem posed by Weinstein and by Kenig, Martel and Robbiano. The talk is based on joint work with E. Lenzmann.

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